where, the first inequality follows by our assumption on the score functions to be bounded, the mean value Theorem [1]} and the fact that \(\left( {a + b} \right)^\lambda \le 2^{\max \left( {\lambda - 1,0} \right)} \left( {a^\lambda + b^\lambda } \right)\)  holds for positive \(a\) , \(b\) , and \(\lambda \) . The second inequality follows by the assumption of \(\varepsilon \)  to be small enough, and the last inequality holds for all \(\varepsilon \)  such that \(\left\Vert  \varepsilon \right\Vert  \le \beta \) . This upper bound is integrable by Holder inequality [1]}, and hence (REF ) is verified.
On the other hand, since \(\mathop {\lim }\limits _{\varepsilon \rightarrow {\bf {0}}} {{o\left( \varepsilon \right)} \mathord {\left\bad.{\vphantom{{o\left( \varepsilon \right)} {\left\Vert  \varepsilon \right\Vert }}} \right.\hspace{0.0pt}} {\left\Vert  \varepsilon \right\Vert }} = 0 \)  and we assume that the
score functions are bounded, (REF ) readily follows. From (REF ) and (REF ) we get the desired result,
\(h\left( {{\textbf {x}} + {\delta }_{\textbf {x}} } \right) - h\left( {\textbf {x}} \right) =- {\mathop {\rm E}\nolimits }\left\lbrace  {{\phi }_{\textbf {x}}^T \left( {{\textbf {x}}} \right){\bf {\tilde{X}\varepsilon }}} \right\rbrace  + o\left( \varepsilon \right)\) 
